A note on “small representations of finite distributive lattices as congruence lattices”

G. Grätzer, H. Lakser, and E. T. Schmidt proved that every distributive lattice with n join-irreducible elements can be represented as the congruence lattice of a small lattice L, that is, a lattice L with O(n ² ) elements. G. Grätzer, I. Rival, and N. Zaguia proved that, for any <2, O(n ² ) can not be improved to O(n ). In this note we show that the theorem about small representation can be improved further to get a more delicate result.     

Separation of polycyclic aromatic hydrocarbons by micellar electrokinetic chromatography with aqueous short-chain alcohol solvent

Micellar electrokinetic capillary chromatography (MEKC) with aqueous organic solvent has been developed to separate polycyclic aromatic hydrocarbons (PAHs). Methanol, ethanol or propanol as an organic modifier was added to sodium dodecyl sulfate (SDS) micellar solution in order to increase the solubility of very hydrophobic solutes in mobile phase. Both methanol and ethanol can be used as co-solvents for the separation of PAHs. Use of ethanol resulted in a shorter analysis time than use of methanol. The separations of some PAHs were unsatisfactory using propanol although the analysis time was much shorter than with ethanol. The influence of ethanol content, SDS concentration and temperature on the separations was studied. Benzene and nine polycyclic aromatic hydrocarbons were successfully separated using 50 mM SDS-20 mM phosphate-5 mM borate, containing 40% (v/v) ethanol at 35 degrees C. The relative standard deviation (R.S.D.) of t(R) ranged from 0.5 to 1.5% for six repeat injections.